OK. You will be able to afford a very decent space for $3,500. Best deal is to find a "trade" with or a sublet from another academic family.
Check out Google maps to make sure you get an apartment near (1) public transport (Red line or #1 bus to MIT) and (2) green space or play space for kids.
Some neighborhoods to consider:
1. Cambridge, Central Square
Areas south of Mass Ave tend to be safer, especially as you approach the Charles River.
Scope out apartments in two families near "Magazine Beach."
2. Cambridge, Kendall Square
Practically on MIT's campus. Look for the more modern construction.
3. Cambridge, Harvard Square
Great place to be!
4. Boston, Back Bay
Get apartment as close as possible to Mass Ave to catch the number 1 bus OR to walk across the Mass Ave bridge to MIT. Best if you stay between Boylston Street and the River.
5. Boston, Beacon Hill/Charles Street
Excellent location, strategically speaking.
Caveat: For safety reasons, I am a tad wary of the Fenway, even if it is a great location and one can catch the #47 bus to MIT. Having lived there for several years, I can tell you that the Red Sox games at Fenway Park are not fun for area residents.
Town west of Boston is very kid friendly and safe. Easy enough to catch bus to Cambridge (#66?).
For the bus, you will want to live as close as Possible to Harvard Street, so find a place near Brookline Village or the Coolidge Corner area
OR Washington Square (Beacon Street) OR Tappan Street/Beacon Street for easiest access to 2 branches of Green Line (green to red takes you to MIT)
When you get here, check out ZIP car for easy, convenient "cars when you need them."
Enjoy your stay.
p.s. You might consider asking at Boston Bed and Breakfast for short term rentals:
0800-89-5128 (Free from U.K.)